A Note on Strong Axiomatization of Gödel Justification Logic

Abstract

Justification Logics are special kinds of modal logics which provide a framework for reasoning about epistemic justification. For this, they extend classical boolean propositional logic by a family of necessity-style modal operators "t :", indexed over t by a corresponding set of justification terms, which thus explicitly encode the justification for the necessity assertion in the syntax. With these operators, one can therefore not only reason about modal effects on propositions but also about dynamics inside the justifications themselves. We replace this classical boolean base with Gödel logic, one of the three most prominent fuzzy logics, i.e. special instances of many-valued logics, taking values in the unit interval [0, 1], which are intended to model inference under vagueness. We extend the canonical possible-world semantics for Justification Logic to this fuzzy realm by considering fuzzy accessibility-and evaluation-functions evaluated over the minimum t-norm and establish strong completeness theorems for various fuzzy analogies of prominent extensions for basic Justification Logic.

Standard Gödel modal logics are not realized by Gödel justification logics

“…We have shown that the four Gödel justification logics GJ CS , GJ T CS , GJ 4 CS , GLP CS from [16] do not realize the standard Gödel modal logics GK , GT , GK4 and GS4 from [3,4] and by this answered one of the open problems in [8,16] negatively. The Gödel justifications logics arise as natural generalizations of the classical cases, both in model theoretic and proof theoretic terms.…”

“…In this paper, we investigate this property for fuzzy variants of modal and justification logics, namely standard Gödel modal logics as introduced by Caicedo and Rodriguez in [3,4] and Gödel justification logics as introduced by Ghari in [8] and Pischke in [16]. These variants replace the classical boolean base of classical justification logic with [0, 1]-valued Gödel logics, one of the three main t-norm based fuzzy logics in the sense of Hájek [11], and initially originating from an intuitionistic perspective along the lines of Gödel [10], Dummett [6] and Horn [12].…”

“…The other standard semantics for classical justification logics defined by so called Fitting-models, see [7], also extends to the fuzzy cases, see e.g. [8,9,16].…”

“…Lemma 1 (Lifting lemma, P. [16]). Let GJ L 0 ∈ {GJ 0 , GJ T 0 , GJ 4 0 , GLP 0 , GJ 45 0 , GJ T 45 0 } and CS be an axiomatically appropriate constant specification for GJ L 0 .…”

Bayesian Confirmation and Justifications

“…Milnikel [19] was the first to investigate uncertain justifications. This lead to several further frameworks that model uncertain reasoning in justification logic: fuzzy justification logics [12,21], possibilistic justification logics [11,28], probabilistic justification logics [13,14,20], and logics for combining evidence and uncertainty [1,25].…”